The paper proposes a novel weighted convolution for signals defined on regular grids (2D images) by optimising a density function that scales the contribution of neighbouring pixels according to their distance from the central pixel. This choice differs from the uniform convolution, which treats all neighbouring pixels equally. Given a convolutional network, we compute the optimal density by minimising a loss functional, with the density function as the variable. Then, the weighted convolution is applied to CNNs to improve the accuracy of image-processing tasks (denoising, classification). The optimal density improves the convergence of the weights of the learning model with respect to the uniform density. The framework separates the optimisation of the convolutional kernel weights (using a stochastic gradient descent method) from the density optimisation (using the DIRECT-L algorithm). Experimental tests on SOTA learning models for image denoising and classification show that the weighted convolution significantly improves the performance compared to standard convolution. For example, the denoising of the DIV2K dataset with Gaussian noise through the DnCNN model with the weighted convolution reaches a PSNR of 31.02, compared to the 29.09 value of the model with the standard convolution. While our method increases execution time by on standard hardware and by on GPU in modern HPC environments, it is robust across several hyperparameters of the learning model. • We introduce an optimal weighted convolution that applies a density function to scale pixel contribution. • A density function is optimised to increase the accuracy of learning models. • Optimal weighted convolution improves image denoising and classification problems.

Alzheimer's disease (AD) is a progressive neurodegenerative disorder and the leading cause of dementia. Magnetic Resonance Imaging (MRI) combined with Machine Learning (ML) enables early diagnosis, but ML models often underperform when trained on small, heterogeneous medical datasets. Transfer Learning (TL) helps mitigate this limitation, yet models pre-trained on 2D natural images still fall short of those trained directly on related 3D MRI data. To address this gap, we introduce an intermediate strategy based on synthetic data generation. Specifically, we propose a conditional Denoising Diffusion Probabilistic Model (DDPM) to synthesise 2D projections (axial, coronal, sagittal) of brain MRI scans across three clinical groups: Cognitively Normal (CN), Mild Cognitive Impairment (MCI), and AD. A total of 9000 synthetic images are used for pre-training 2D models, which are subsequently extended to 3D via axial, coronal, and sagittal convolutions and fine-tuned on real-world small datasets. Our method achieves 91.3% accuracy in binary (CN vs. AD) and 74.5% in three-class (CN/MCI/AD) classification on the 3T ADNI dataset, outperforming both models trained from scratch and those pre-trained on ImageNet. Our 2D ADnet achieved state-of-the-art performance on OASIS-2 (59.3% accuracy, 57.6% F1), surpassing all competitor models and confirming the robustness of synthetic data pre-training. These results show synthetic diffusion-based pre-training as a promising bridge between natural image TL and medical MRI data.

Abstract Purpose: Several applications require the super-resolution of noisy images and the preservation of geometrical and texture features. State-of-the-art super-resolution methods do not account for noise and generally enhance the output image’s artefacts (e.g., aliasing, blurring). Methods: We propose a learning-based method that accounts for the presence of noise and preserves the properties of the input image, as measured by quantitative metrics, e.g., normalised crossed correlation, normalised mean squared error, peak-signal-to-noise-ration, structural similarity feature-based similarity, universal image quality. We train our network to up-sample a low-resolution noisy image while preserving its properties. We perform our tests on the Cineca Marconi100 cluster, at the 26th position in the “top500” list. Results: The experimental results show that our method outperforms learning-based methods, has comparable results with standard methods, preserves the properties of the input image as contours, brightness, and textures, and reduces the artefacts. As average quantitative metrics, our approach has a PSNR value of 23.81 on the super-resolution of Gaussian noise images with a 2X up-sampling factor. In contrast, previous work has a PSNR value of 23.09 (standard method) and 21.78 (learning-based method). Conclusion: Our learning-based and quality-preserving super-resolution improves the high-resolution prediction of noisy images with respect to state-of-the-art methods with different noise types and up-sampling factors.

The F-transform has proven to be effective in various applications, such as time series analysis, numerical solutions of differential equations, and signal or image processing. The most important parameter of the F-transform is a fuzzy partition initially introduced for 1-D spaces, with a few generalizations to higher dimensional spaces. However, these generalizations have a limited application to signals defined on domains with arbitrary geometry. To overcome this limitation, we propose using nonseparable membership functions induced by kernels, allowing the application of the F-transform to more general domains. We introduce a universal concept of a fuzzy partition that includes a kernel representation, a fuzzy partition, and the corresponding F-transform. In addition, we discuss the main properties of this generalized F-transform and characterize the nonlocal Laplace operator in terms of the F-transform. We also discuss image denoising as the main application and compare our results with state-of-the-art methods and different noise types and intensities.

The definition of the F-transform has been limited mainly to 1D signals and 2D data due to the difficulty of defining membership functions, their centres, and support on a domain with arbitrary dimensionality and topology. We propose a novel method for the adaptive selection of the optimal centres and supports of a class of radial membership functions based on minimising the reconstruction error of the input signal as the Ftransform and its inverse, or as a weighted linear combination of the membership functions. Replacing uniformly sampled centres of the membership functions with adaptive centres and fixed supports with adaptive supports allows us to preserve the input signal's local and global features and achieve a good approximation accuracy with fewer membership functions. We compare our method with uniform sampling and previous work. As a result, we improve the image reconstruction with respect to compared methods and we reduce the underlying computational cost and storage overhead. Finally, our approach applies to any class of continuous membership functions.

This article addresses the definition and computation of the out-of-sample membership functions and the resulting out-of-sample fuzzy transform (FT), which extend their discrete counterparts to the continuous case. Through the out-of-sample FT, we introduce a coherent analysis of the discrete and continuous FTs, which is applied to extrapolate the behavior of the FT on new data and to achieve an accurate approximation of the continuous FT of signals on arbitrary data. To this end, we apply either an approximated approach, which considers the link between integral kernels and the spectrum of the corresponding Gram matrix, or an interpolation of the discrete kernel eigenfunctions with radial basis functions. In this setting, we show the generality of the proposed approach to the input data (e.g., graphs, 3-D domains) and signal reconstruction.

Data are represented as graphs in a wide range of applications, such as Computer Vision (e.g., images) and Graphics (e.g., 3D meshes), network analysis (e.g., social networks), and bio-informatics (e.g., molecules). In this context, our overall goal is the definition of novel Fourier-based and graph filters induced by rational polynomials for graph processing, which generalise polynomial filters and the Fourier transform to non-euclidean domains. For the efficient evaluation of discrete spectral Fourier-based and wavelet operators, we introduce a spectrum-free approach, which requires the solution of a small set of sparse, symmetric, well-conditioned linear systems and is oblivious of the evaluation of the Laplacian or kernel spectrum. Approximating arbitrary graph filters with rational polynomials provides a more accurate and numerically stable alternative with respect to polynomials. To achieve these goals, we also study the link between spectral operators, wavelets, and filtered convolution with integral operators induced by spectral kernels. According to our tests, main advantages of the proposed approach are (i) its generality with respect to the input data (e.g., graphs, 3D shapes), applications (e.g., signal reconstruction and smoothing, shape correspondence), and filters (e.g., polynomial, rational polynomial), and (ii) a spectrum-free computation with a generally low computational cost and storage overhead.

Abstract Depth sensors of low-cost acquisition devices (e.g. Microsoft Kinect, Asus Xtion) are coming into widespread use; however, 3D acquired data are generally large, heterogeneous, and complex to analyse and interpret. In this context, our overall goal is the analysis of the action of a subject in a 3D video, e.g. the action of a human or the movement of its subparts. To this end, the action classification is achieved through the analysis of the temporal variation of geometric (e.g. centroid path, volume variation, activated voxels) and kinematic (e.g. speed) properties in consecutive frames. Then, these descriptors and the corresponding histograms are used to search a frame in a 3D video and to compare 3D videos. Our approach is applied to 3D videos represented as triangle meshes or point sets, and eventually to an underlying skeleton or to markers (if available). Our tests on the MIT, Berkley, i3DPost, NTU, and DUTH data sets confirm the usefulness of the proposed approach for the analysis and comparison of 3D videos, as well as for action classification.

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.

The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.

Abstract In this paper, we propose a new construction for the Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration from the well‐established methodology of diffusion wavelets. This novel construction allows us to rapidly compute a multi‐scale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this leads to a family of functions that inherit many attractive properties of the heat kernel (e.g. local support, ability to recover isometries from a single point, efficient computation). Due to its natural ability to encode high‐frequency details on a shape, the proposed method reconstructs and transfers ‐functions more accurately than the Laplace‐Beltrami eigenfunction basis and other related bases. Finally, we apply our method to the challenging problems of partial and large‐scale shape matching. An extensive comparison to the state‐of‐the‐art shows that it is comparable in performance, while both simpler and much faster than competing approaches.

The fuzzy transform (F-transform) is ubiquitous in different research fields and applications, such as image and data compression, data mining, knowledge discovery, and the analysis of linguistic expressions. As a generalization of the F-transform, in this article, we introduce the continuous F-transform and its inverse, as an integral operator induced by a kernel function. Through the relation between membership functions and integral kernels, we show that the main properties (e.g., continuity and symmetry) of the membership functions are inherited by the continuous F-transform. Then, the relation between the continuous F-transform and integral operators is used to introduce a data-driven F-transform, which encodes intrinsic information (e.g., structure, geometry, and sampling density) about the input data. In this way, we avoid coarse fuzzy partitions, which group data into large clusters that do not adapt to their local behavior, or a too dense fuzzy partition, which generally has cells that are not covered by the data, thus being redundant and resulting in a higher computational cost. To this end, the data-driven membership functions are defined by properly filtering the spectrum of the Laplace–Beltrami operator associated with the input data. Finally, we introduce the space of continuous F-transforms, which is useful for the comparison of different continuous F-transforms and for their efficient computation.

Abstract This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties (e.g. smoothness, locality, invariance to shape transformations) that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commute‐time distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero. Instead of applying a truncated spectral approximation or prolongation operators, we propose a computation of Laplacian distances and kernels through the solution of sparse linear systems. Our approach is free of user‐defined parameters, overcomes the evaluation of the Laplacian spectrum and guarantees a higher approximation accuracy than previous work.

Abstract In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute‐time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace‐Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute‐time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application.

Nowadays, a wide range of advanced techniques provides accurate and detailed 3D data about patients' anatomy, as captured by medical scans (MRI, CT, Micro CT, etc.). While medical imaging assists daily clinical practice, 3D patient-specific models (3D-PSMs) of anatomy have still a quite limited use. We consider part-based semantic annotation beneficial to bring 3D-PSMs into clinical practice. To this end, tools are needed to extract clinically relevant information from 3D models, to associate such knowledge with their corresponding parts, and to support the storage, sharing and searching of annotated 3D-PSMs in a structured manner. In this context, we present the Sem Anatomy3D framework, which demonstrates the idea of ontology-driven annotation and indexing of 3D-PSMs and their Parts-of-Relevance, characterized by anatomical landmarks and pathological markers (e.g. Articular and non-articular facets, ligament insertion sites, erosions). The key functionaity is to offer services for part-base annotation of 3D-PSMs which enables search or browse the 3D-PSM according to the annotation attached to its Parts-of-Relevance. The paper describes the results in terms of methods to support the part-based annotation of 3D-PSM, and the formalization of the data model to store and manage global and part-based annotation to improve search and analysis of 3D patient-specific anatomical models and subparts. Finally, we specialized our framework to support the diagnosis of rheumatoid arthritis in the carpal bones, but, in principle, it can support similar tasks in other clinical applications.

The exploration of biomedical data which involves heterogeneous sources coming from different spatial scales and medical domains is a challenging topic in current research. In this work, we combine efforts regarding multi-scale visualization, multimodal interaction and knowledge formalization for the exploration of multi-scale biomedical data. The knowledge formalization stores and organizes the information sources, the integrated visualization captures all relevant information for the domain expertise of the user and the multimodal interaction provides a natural exploration. We present a concrete example of use of the proposed exploratory system designed for a biologist investigating multi-scale pathologies.

Barycentric coordinates yield a powerful and yet simple paradigm to interpolate data values on polyhedral domains. They represent interior points of the domain as an affine combination of a set of control points, defining an interpolation scheme for any function defined on a set of control points. Numerous barycentric coordinate schemes have been proposed satisfying a large variety of properties. However, they typically define interpolation as a combination ofallcontrol points. Thus alocalchange in the value at a single control point will create aglobalchange by propagation into the whole domain. In this context, we present a family oflocal barycentric coordinates(LBC), which select for each interior point a small set of control points and satisfy common requirements on barycentric coordinates, such as linearity, non-negativity, and smoothness. LBC are achieved through a convex optimization based on total variation, and provide a compact representation that reduces memory footprint and allows for fast deformations. Our experiments show that LBC provide more local and finer control on shape deformation than previous approaches, and lead to more intuitive deformation results.

Recent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes.

In digital geometry processing and shape modeling, the Laplace-Beltrami and the heat diffusion operator, together with the corresponding Laplacian eigenmaps, harmonic and geometry-aware functions, have been used in several applications, which range from surface parameterization, deformation, and compression to segmentation, clustering, and comparison. Using the linear FEM approximation of the Laplace-Beltrami operator, we derive a discrete heat kernel that is linear, stable to an irregular sampling density of the input surface, and scale covariant. With respect to previous work, this last property makes the kernel particularly suitable for shape analysis and comparison; in fact, local and global changes of the surface correspond to a re-scaling of the time parameter without affecting its spectral component. Finally, we study the scale spaces that are induced by the proposed heat kernel and exploited to provide a multi-scale approximation of scalar functions defined on 3D shapes.

Abstract We focus on the class of ‘regular’ models defined by Várady et al. for reverse engineering purposes. Given a 3D surface represented through a dense set of points, we present a novel algorithm that converts to a hierarchical representation . In , the surface is encoded through patches of various shape and size, which form a hierarchical atlas. If belongs to the class of regular models, then captures the most significant features of at all the levels of detail. In this case, we show that can be exploited to interactively select regions of interest on and intuitively re‐design the model. Furthermore, intrinsically encodes a hierarchy of useful ‘segmentations’ of . We present a simple though efficient approach to extract and optimize such segmentations, and we show how they can be used to approximate the input point sets through idealized manifold meshes.

We present the results of a project aimed to evaluate 2D and 3D face recognition algorithms. In particular, we focused on the potentialities of 3D-based techniques to overcome typical limitations of 2D methods in non-controlled situations. According to the reference scenario of people identification at airport check points, we built a representative database on which we tested different face recognition algorithms. We implemented and tested an improved version of a well-known state-of-the-art 3D approach, and verified that on our dataset it performs better than a widely used commercial system.

The behavior of a variety of phenomena measurable on the boundary of 3D shapes is studied by modeling the set of known measurements as a scalar functionf:P → R, defined on a surface P. Furthermore, the large amount of scientific data calls for efficient techniques to correlate, describe, and analyze this data. In this context, we focus on the problem of extending the measures captured by a scalar functionf, defined on the boundary surface P of a 3D shape, to its surrounding volume. This goal is achieved by computing a sequence of volumetric functions that approximatefup to a specified accuracy and preserve its critical points. More precisely, we compute a smooth mapg: R3→ R such that the piecewise linear functionh:=gP : P → R, which interpolates the values ofgat the vertices of the triangulated surface P, is an approximation offwith the same critical points. In this way, we overcome the limitation of traditional approaches to function approximation, which are mainly based on a numerical error estimation and do not provide measurements of the topological and geometric features off. The proposed approximation scheme builds on the properties offrelated to itsglobal structure, that is, its critical points, and ignores the local details off, which can be successively introduced according to the target approximation accuracy.

Given a manifold surface {\cal M} and a continuous scalar function f:{\cal M}\rightarrow {\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}, the Reeb graph of ({\cal M},f) is a widely used high-level descriptor of {\cal M} and its usefulness has been demonstrated for a variety of applications, which range from shape parameterization and abstraction to deformation and comparison. In this context, we propose a novel contouring algorithm for the construction of a discrete Reeb graph with a minimal number of nodes, which correspond to the critical points of f (i.e., minima, maxima, and saddle points) and its level sets passing through the saddle points. In this way, we do not need to sample, sweep, or increasingly sort the f-values. Since most of the computation uses only local information on the mesh connectivity, equipped with the f-values at the surface vertices, the proposed approach is insensitive to noise and requires a small-memory footprint and temporary data structures. Furthermore, we maintain the parametric nature of the Reeb graph with respect to the input scalar function and we efficiently extract the Reeb graph of time-varying maps. Indicating with n and s the number of vertices of {\cal M} and saddle points of f, the overall computational cost O(sn) is competitive with respect to the O(n\,\log \,n) cost of previous work. This cost becomes optimal if {\cal M} is highly sampled or s\le \log n, as it happens for Laplacian eigenfunctions, harmonic maps, and one-forms.

Given a manifold surface M and a continuous function f : M rarr R, the Reeb graph of (M, f) is a widely-used high-level descriptor of M and its usefulness has been demonstrated for a variety of applications, which range from shape parameterization and abstraction to deformation and comparison. In this context, we propose a novel computation of the Reeb graph that is based on the analysis of the iso-contours solely at saddle points and does not require sampling or sweeping the image of f. Furthermore, the proposed approach does not use global sorting steps of the function values and exploits only a local information on f, without handling it as a whole. By combining the minimal number of nodes in the Reeb graph with the use of a small amount of memory footprint and temporary data structures, the overall computation takes O(sn)-time, where n is the number of vertices of the triangulation of M and s is the number of saddles of f. Finally, the technique can be easily extended to compute the Reeb graphs of time-varying functions.

In this paper we propose a method for the automatic extraction and annotation of the animation control skeleton of virtual humans, which relies on an a-priori knowledge of the human anatomy. The method is based on a segmentation of the virtual human shape into semantically meaningful features, like arms or legs, and on an automatic location and labeling of joints of the control skeleton. The method is particularly relevant for computer animation where the process still largely relies on manual tasks, and especially for virtual characters built on real scanned data. Several examples will show the results obtained with our approach.

Recent advances in the parameterization and adaptive sampling of disc-like surfaces have brought a renewed interest on the global parameterization problem and, more specifically, on the cut-graph search. This paper focuses on the calculation of a family of generators and cut-graphs for the global parameterization of arbitrary triangle meshes. This result is achieved by combining the construction of harmonic scalar fields f : M rarr R of known maxima and minima with the quasi Morse-Smale complex of(M, f). The proposed technique has a simple implementation and outperforms previous work in terms of smoothness of the cut-graphs, stability with respect to the surface sampling, tessellation, topological noise (e.g., tiny handles), and capability of handling boundary components. Since we generate a family of cut-graphs, we also provide a comparison between the parameterizations of M induced by two cut-graphs.

Mesh segmentation has become an important component in many applications in computer graphics. In the last several years, many algorithms have been proposed in this growing area, offering a diversity of methods and various evaluation criteria. This paper provides a comparative study of some of the latest algorithms and results, along several axes. We evaluate only algorithms whose code is available to us, and thus it is not a comprehensive study. Yet, it sheds some light on the vital properties of the methods and on the challenges that future algorithms should face

3D shape approximation and processing with implicitly defined surface primitives (radial basis functions and more general kernel-based approximations, partition of unity approximations, moving least squares, etc.) is currently a subject of intensive research in geometric modeling and computer graphics. In this paper, we propose an approach that combines two conflicting criteria: achieving high approximation accuracy and obtaining an economical surface representation. We employ compactly supported radial basis functions and use Tikhonov regularization to achieve a near optimal selection of their centers. An iterative approach, which defines a multi-level approximation, is used to cope with arising constrained optimization problems

Abstract This paper describes a novel approach to the parameterization of triangle meshes representing 2‐manifolds with an arbitrary genus. A topology‐based decomposition of the shape is computed and used to segment the shape into primitives, which define a chart decomposition of the mesh. Then, each chart is parameterized using an extension of the barycentric coordinates method. The charts are all 0‐genus and can be of three types only, depending on the number of boundary components. The chart decomposition and the parameterization are used to define a shape graph where each node represents one primitive and the arcs code the adjacency relationships between the primitives. Conical and cylindrical primitives are coded together with their skeletal lines that are computed from and aligned with their parameterization. The application of the parameterization approach to remeshing guarantees that extraordinary vertices are localized only where two patches share a boundary and they are not scattered on the whole surface.

The paper investigates the main issues related to the definition of abstraction tools for deriving high-level descriptions of complex geometric models. Among the wide range of shape descriptors, topological graph-like representations not only give a powerful and synthetic sketch of the object, but also capture its inner structure, that is how features connect together to give the overall shape. This aspect makes them useful to describe complex 3D objects in various applications like modeling, morphing, matching and recognition. The paper surveys the main properties of skeletons developed in shape modeling for representing objects.

Different application fields have shown increasing interest in shape description oriented to recognition and similarity issues. Beyond the application aims, the capability of handling details separating them from building elements, the invariance to a set of geometric transformations, the uniqueness and stability to noise represent fundamental properties of each proposed model. This paper defines an affine-invariant skeletal representation; starting from global features of a 3D shape, located by curvature properties, a Reeb graph is defined using the topological distance as a quotient function. If the mesh has uniformly spaced vertices, this Reeb graph can also be rendered as a geometric skeleton defined by the barycenters of pseudo-geodesic circles sequentially expanded from all the feature points.